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CHAPTER 1
INTRODUCTION
1.1 History
For nearly 150 years, the known fundamental passive circuit elements were limited to the
capacitor (discovered in 1745), the resistor (1827), and the inductor (1831). Then, in a
brilliant but underappreciated 1971 paper, Leon Chua, a professor of electrical engineering at the
University of California, Berkeley, predicted the existence of a fourth fundamental device, which
he called a memristor. He proved that memristor behavior could not be duplicated by any
circuit b u i l t
using
only
the
other three
elements,
which is why
the
m e m r i s t o r i s t r u l y fundamental.
Memristor is a contraction of “memory resistor,” because that is exactly its function to
remember its history. A memristor is a two-terminal device whose resistance depends on the
magnitude and polarity of the voltage applied to it and the length of time that voltage has been
applied. When you turn off the voltage, the memristor remembers it’s most recent resistance until
the next time you turn it on, whether that happens a day later or a year later.
Chua discovered a missing link in the pair wise mathematical equations that relate the
four circuit quantities-charge, current, voltage, and magnetic flux-to one another. These can be
related in six ways. Two are connected through the basic physical laws of electricity and
magnetism, and three are related by the known circuit elements: resistors connect voltage and
current, inductors connect flux and current, and capacitors connect voltage and charge. But one
equation is missing from this group: the relationship between charge moving through a circuit
and the magnetic flux surrounded by that circuit.
Chua demonstrated mathematically that his hypothetical device would provide a
relationship between flux and charge similar to what a nonlinear resistor provides between
voltage and cur- rent. In practice, that would mean the device’s resistance would vary according
to the amount of charge that passed through it. And it would remember that resistance value
even after the current was turned off.
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1.2 Theory
Fig 1.2.1
The memristor is formally defined as a two-terminal element in which the
magnetic flux Φm between the terminals is a function of the amount of electric charge q that
has passed through the device. Each memristor is characterized by its memristance function
describing the charge-dependent rate of change of flux with charge.
Noting from Faraday's law of induction that magnetic flux is simply the time integral of
voltage, and charge is the time integral of current, we may write the more convenient form.
It can be inferred from this that memristance is simply charge-dependent resistance.
If M (q (t)) is a constant, then we obtain Ohm's Law R (t) =V (t)/ I (t). If M (q (t)) is
nontrivial, however, the equation is not equivalent because q (t) and M (q (t)) will vary with time.
Solving for voltage as a function of time we obtain
V (t) = M (q (t)) I (t)
This equation reveals memristance defines a linear relationship between current and
voltage, as long as M does not vary with charge. Of course, nonzero current implies time varying
charge.
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Alternating current, however, may reveal the linear dependence in circuit operation by
inducing a measurable voltage without net charge movement as long as the maximum change in
q does not cause much change in M. Furthermore, the memristor is static if no current is applied.
If I (t) = 0, we find V (t) = 0 and M (t) is constant. This is the essence of the memory effect.
The power consumption characteristic recalls that of a resistor, 𝐼 2 R.
As long asM (q(t )) varies little, such as under alternating current, the memristor will appear as a
resistor. If M (q(t )) increases rapidly, however, current and power consumption will quickly
stop.
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CHAPTER2
DETAILED DISCRIPTION
2.1 Structure of Titanium Diode Memristor
Fig 2.1.1
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The HP device is composed of a thin (50 nm) titanium dioxide film between two 5nm
thick electrodes, two platinum wires. Initially, there are two layers to the titanium dioxide film,
TiO2 and TiO2- x. The upper layer
The oxygen
vacancies
the vacancies themselves
has
a slight
are donors
positively charged.
depletion
of
of electrons
Stoichiometric
oxygen
which
TiO2 act
atoms.
makes
as a
insulator. (It is a semiconductor) but oxygen deficient TiO2-x is a conductor and have lower
resistance than the stoichiometric compound.
2.2 Working
Fig 2.2.1
If a positive voltage is applied to the top electrode of the device, it will repel the (also
positive)oxygen vacancies in the TiO2-xlayer down into the pure TiO2 layer. That turns the
TiO2 layer into TiO2- x and makes it conductive, thus turning the device on. A negative voltage
has the opposite effect: the vacancies are attracted upward and back out of the TiO2, and thus the
thick- ness of the TiO2 layer increases and the device turns off.
The oxygen deficiencies in the TiO2-x manifest as “bubbles” of oxygen vacancies
scattered throughout the upper layer. A positive voltage on the switch repels
the (positive)oxygen deficiencies in the metallic upper TiO2- x layer, sending them into the
insulating TiO2layer below. That causes the boundary between the two materials to move down,
increasing the percentage of conducting TiO2- x and thus the conductivity of the entire
switch. The more positive voltage is applied, the more conductive the cube becomes.
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A negative voltage on the switch attracts the positively charged oxygen bubbles, pulling
them out of the TiO2. The amount of insulating, resistive TiO2 increases, thereby making the
switch as a whole resistive. The more negative voltage is applied, the less conductive the
cube becomes. What makes this switch special is that when the voltage is turned off,
positive or negative, the oxygen bubbles do not migrate. They stay where they are,
which means that the boundary between the two titanium dioxide layers is frozen.
That is
how the
memristor “remembers”
how
much
voltage
was
last
applied.
Resistance also depends on the length of time that voltage has been applied.
Fig 2.2.2
A memristor’s structure, shown here in a scanning tunneling microscope
image, will enable dense, stable computer memories.
 Bow Ties:
Leon Chua’s original graph of the hypothetical memristor’s behavior is shown at top right;
the graph of R. Stanley William’s experimental results are shown below. The loops
map the switching behavior of the device: it begins with a high resistance, and as the voltage
increases, the current slowly increases. As charge flows through the device, the resistance drops,
and the current increases more rapidly with increasing voltage until the maximum is reached.
Then, as the voltage decreases, the current decreases but more slowly, because charge is flowing
through the device and the resistance is still dropping. The result is an on-switching loop. When
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the voltage turns negative, the resistance of the device increases, resulting in an off-switching
loop.
Fig 2.2.3
2.3 Other Types of Memristors
 Spintronic Memristor
Concept of Spintronic memristor is given as, resistance is caused by the spin of electrons
in one section of the device pointing in a different direction than those in another section,
creating a "domain wall," a boundary between the two states. Electrons flowing into
the device have a certain spin, which alters the magnetization state of the device.
Changing the magnetization, in turn, moves the domain wall and changes the device's resistance.
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 Spin Torque Magnetoresistance
Spin Torque Transfer MRAM is a well-known device that exhibits memristive behavior.
The resistance is dependent on the relative spin orientation between two sides of
a magnetic tunnel junction. This in turn can be controlled by the spin torque induced by the
current flowing through the junction.
However, the length of time the current flows through the junction determines
the amount of current needed, i.e., the charge flowing through is the key variable. Additionally,
MgO based magnetic tunnel junctions show memristive behavior based on the drift of
oxygen vacancies within the insulating MgO layer (resistive switching). Therefore,
the combination of spin transfer torque and resistive switching leads naturally
to a second-order memristive system with w=(w1,w2) where w1 describes the magnetic state
of the magnetic tunnel junction and w2 denotes the resistive state of the MgO barrier.
Note that in this case the change of w1 is current-controlled (spin torque is due to a high
current density) whereas the change of w2 is voltage-controlled (the drift of oxygen vacancies is
due to high electric fields).
 Polymeric Memristor
Juri H. Krieger and Stuart M. Spitzer claim to have developed a polymeric
memristor
before
the
titanium
dioxide
memristor
more
recently
announced.
T h e i r w o r k d e s c r i b e s t h e p r o c e s s o f d y n a m i c d o p i n g o f p o l ym e r a n d i n o r g a
n i c dielectric-like materials in order to improve the switching characteristics and retention
required to create functioning nonvolatile memory cells. Described is the use of a special passive
layer between electrode and active thin films, which enhances the extraction of ions
from the electrode. It is possible to use fast ion conductor as this passive layer,
which allows to significantly decreasing the ionic extraction field.
 Resonant Tunnelling Diode Memristor
1994, F. A. Buot and A. K. Rajagopal demonstrated that a 'bow-tie' current-voltage (I-V)
characteristics occurs in AlAs/GaAs/AlAs quantum-well diodes containing special doping design
of the spacer layers in the source and drain regions, in agreement with the published
experimental results This 'bow-tie' current-voltage (I-V) characteristic is sine qua non of a
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memristor although the term memristor is not explicitly mentioned in their papers. No magnetic
interaction is involved in the analysis of the 'bow-tie' I-V characteristics.
 3-Terminal Memristor
Although the memristor is defined in terms of a 2-terminal circuit element, there was an
implementation of a 3-terminal device called a memistor developed by Bernard Widrow in
1960.Memistors formed basic components of a neural network architecture
called ADALINE developed by Widrow
and Ted Hoff the memistor was described as
follows:
Like the transistor, the memistor is a 3-terminal element. The conductance between two
of t h e
terminals
is controlled
by the
time
integral of
the current in
t h e t h i r d , r a t h e r t h a n i t s instantaneous value as in the transistor.
Reproducible
elements
have been
made which
are
continuously variable
(thousands of possible analog storage levels), and which typically vary in resistance from 100
ohms to 1 ohm, and cover this range in about 10 seconds with several mille amperes of plating
current.
Adaptation is accomplished by direct current while sensing the neuron logical structure is
accomplished nondestructively by passing alternating currents through the arrays of memistor
cells.
Since the conductance was described as being controlled by the time integral of current as
in Chua's theory of the memristor, the memistor of Widrow may be considered as a form
of memristor having three instead of two terminals.
However, one of the main limitations of Widrow's memistor was that they were
made from an electroplating cell rather than as a solid-state circuit element. Solid-state circuit
elements were required to achieve the scalability of the integrated circuit which was gaining
popularity around the same time as the invention of memristor Windrow’s memristor.
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2.4Array Based Multilevel Memory of Memristor
The proposed method has the operating point of the memristor be maintained its desired
location (or resistance value) utilizing a set of pre-determined multiple resistance levels.
Fig.2.4.1 shows the basic idea of the proposed method, where the resistance array to be
referenced and the memristor to be programmed (tuned) are shown.
The goal is to have the memristor keep any of the resistance level selected from the
resistance array. If a predetermined magnitude of the current pulse Is (t) is applied to the
resistance array, different levels of voltages V k will appear at each node of the resistance array.
The same current pulse is (t) is also applied to the memristor.
Fig 2.4.1
The programming (tuning) of the memristor is performed by applying additional
current pulses to the memristor with the appropriate directions until the voltage of the memristor
equals to that of the selected node voltage in the resistance array. If the voltage of the memristor
reaches that of the selected node, the resistance value of the memristor becomes the same as the
partial sum of the resistance from the ground to the selected node of the resistance array.
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This idea is employed in both the “write-in” and the “read-out/restoration” circuits.
Detailed description of these circuits will be presented in the following sections.
2.5 Memristor Write-In Circuit
Fig 2.5.1
The memristor write-in circuit is used to bias the memristor at a desired resistance level.
The critical write-in circuit is shown in Fig 2.5.1.
The first step is to choose the write-in memristor and the resistance value to be
memorized by turning on one of the switches in switch array S1 of Fig2.5.1.
and the
corresponding switch pair in switch array S4 respectively.
Then, an initial current pulse I s(t) is applied at the drain of the transistor Q1
so that its mirrored current pulses appear at transistors Q2 and Q3. With this current pulse,
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negative voltages appear at both the selected reference nodes and at the output terminal V out of
the memristors.
Suppose the selected memristance M j is less than the referenced sum of the resistances
Rk sum , in this particular case, Diff k+ is smaller than Diff k- since Vout (Tp) is less negative
than that of Vk (Tp). These Diff k outputs caused the comparator C1 to generate
a positive pulse. Note that the negative and the positive output terminals of Diff k are linked to
the positive and the negative input terminals of C1 respectively.
As a consequence, switch S3 is turned on. Ø such increased flux Ø, the increment of the
memristance can be obtained with a monotonically increasing function via the R vs. Ø
graph in Fig 3.2.3 As a consequence, the memristor voltage decreases toward the selected
reference level.
The processing from the above voltage difference computation repeats until
the difference between Vk (Tp) and Vout (Tp) becomes zero, thereby completing the
“write-in” processing of the reference resistance Rk sum.
On the other hand, when the selected memristance M j is larger than the referenced sum
Rk sum of the resistances, the memristance of the selected memristor is decreased and the
memristor voltage is increased toward the selected reference level through the opposite
procedure mentioned above.
The above comparison between the voltages and the adjustment of the memristance are
repeated until the memristor voltage is equal to its selected reference voltage level.
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2.6 Memristor Read-Out/Restoration Circuit
Fig 2.6.1
The memristor read-out/ restoration circuit is used to read the content of the memristor by
applying an appropriate integrating current or voltage. The critical function of this
circuit is to guarantee the memristor will stay at a set of fixed values without being perturbed
when a read- out voltage or a noise voltage is applied across the memristor. To achieve this goal,
a single compensating pulse is applied to have the memristance changed toward the closest
reference resistance after the initial read-out pulse is applied.
The read-out circuit is the same as the write- in circuit except the negative signal
excluding circuit (N_Excld), MIN A and MIN B circuits as shown in Fig.2.6.1. The N_Excld is
the circuit to choose only the positive signals from Diffk+ or Diff k- using the negative
signal excluding circuit N_Excld by comparing between the DC voltage and the
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output of the Diff circuit. The circuits MIN A and MIN B together with the comparator
C1
are used to choose the smallest absolute value among all Diff k+ and Diff k- signals.
If the output of MIN A is smaller than that of MIN B, the memristor voltage is
higher than that of its closest reference voltage (with M j< Rk sum).
In this case, the memristance M j should be increased. On the other hand, if
the output of the MIN A is larger than that of MIN B, then the memristor voltage is smaller
than that of its closest reference voltage(with M j>Rk sum). In this case, M j should be
decreased.
The above adjustment of the memristor is executed only once during each read-out
processing.
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CHAPTER 3
ADVANTAGES AND DISADVANTAGES
3.1Advantages
When you turn off the voltage, the memristor remembers its most recent resistance until
the next time you turn it on, whether that happens a day later or a year later. This freezing
property suits memristors brilliantly for computer memory. The ability to indefinitely store
resistance values means that a memristor can be used as a nonvolatile memory. That might not
sound like very much, but go ahead and pop the battery out of your laptop, right now—no
saving, no quitting, nothing. You’d lose your work, of course. But if your laptop were built
using a memory based on memristors, when you popped th e battery back in, your
screen would return to life with everything exactly as you left it: no lengthy reboot, no
half-
dozen auto-recovered files.
There are several advantages of the memristor memory over conventional transistorbased memories. One is its strikingly small size. Though memristor is still at its
early development stage, its size is at most one tenths of its RAM counterparts. If
the fabrication technology for memristor is improved, the size and advantage could be even
more significant .
Another feature of the memristor is its incomparable potential to store analog information
which enables the memristor to keep multiple bits of information in a memory cell.
Besides
these
features,
the memristor
is also
an ideal
device
for implementing synaptic weights in artificial neural networks. Williams' solid-state
memristors can be combined into devices called crossbar latches, which could replace transistors
in future computers, taking up a much smaller area.
They
can also
be fashioned
into non-volatile solid-state memory,
which would allow greater data density than hard drives with access times potentially similar
to DRAM, replacing both components HP prototyped a crossbar latch memory using
the devices that can fit 100gigabits in a square centimeter, and has designed a highly
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scalable 3D design (consisting of up to1000 layers or 1 petabit in a cubic CM) has
reported that its version of the memristor is currently about one-tenth the speed
of DRAM . The devices' resistance would be read with alternating current so that they do
not affect the stored value.
3.2disadvantages
Despite many favorable features, memristors have several weaknesses in practice. One
weakness comes from the nonlinearity in the Ø vs. q curve which makes it difficult to determine
the proper pulse width for achieving a desired resistance value.
If the nonlinearity is spatially variant in the die of a chip which is common in the
fabrication process, the difficulty could be very serious. Another difficulty comes from the
property of the memristor which integrates any kind of signals including noise that appeared at
the memristor and results in the memristors being perturbed from its original pre-set values.
Thus, the resistance can be interpreted as the slope at an operating point on the Ø-q curve.
If the Ø-q curve is nonlinear, the resistance will vary with the operating point. For instance, if the
Ø-q curve is the nonlinear function Shown in Fig. 3.2.1, its small-signal resistance can be
obtained by re-plotting it as a function of Øq in the R vs .Ø plane as in Fig. 3.2.2.Since the flux
Ø is obtained by integrating the voltage, the resistance of the memristor can be controlled by
applying a voltage signal across the memristor.
The principle of the memristor is based on the nonlinear property of basic circuit
elements. In the relationships defining basic circuit elements, charge is defined as the time
integral of current, namely,
Fig 3.2.1
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Fig 3.2.2
17
Fig 3.2.3
The above memristance tuning method assumes an ideal operating condition.
In practice, there are some problems that must be overcome. The first problem is caused by then
nonlinearitybetween the applied voltage andthe corresponding resistance. Suppose the
resistance characteristics of the memristors is different from each other as shown in Fig. 3.2.3,where
the resistance between the applied voltage and the corresponding resistance. Suppose the
resistance characteristics of the memristors is different from each other as shown in Fig. 3.2.3,where
the resistance R d is obtained at different values of Ø such as Ø1Ø2 and Ø3. If the same
magnitude of voltage pulses is chosen, then the durations of the pulse widths for obtaining
the same resistance will be different depending on the characteristics of the memristors.
Another problem comes from the fact that the operating point and its
associated memristance would be changed whenever some voltage is applied across the
memristor. The voltage applied for read-out or even noise voltages would be integrated which
causes the flux Ø to be altered. Again, this causes the programmed resistance to be varied. Chua
had suggestedapplying a voltage doublet with equal positive and negative read-out pulses to
resolve such problem. However, the problem remains if the positive and the negative
pulses are not perfectly identical due to the non-ideal pulse-generation circuits.
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CHAPTER 4
FUTURE SCOPE
Combined with transistors in a hybrid chip, memristors could radically improve
the performance of digital circuits without shrinking transistors. Using transistors more
efficiently could in turn give us another decade, at least, of Moore’s Law performance
improvement, without requiring the costly and increasingly difficult doublings of transistor
density on chips. In the end, memristors might even become the cornerstone of new analog
circuits that compute using architecture much like that of the brain. Memristors potential goes far
beyond instant-on computers to embrace one of the grandest technology challenges: mimicking
the functions of a brain. Within a decade, memristors could
let us emulate, instead of merely
simulate, networks of neurons and synapses. Many research groups have been working
toward a brain in silico: IBM’s Blue Brain project, Howard Hughes Medical Institute’s
Janelia Farm, and Harvard’s Center for Brain Science are just three. However, even a mouse
brain simulation in real time involves solving an astronomical number of coupled partial
differential equations. A digital computer capable of coping with this staggering workload would
need to be the size of a small city, and powering it would require several dedicated nuclear
power plants.
Memristors can be made extremely small, and they function like synapses. Using them,
we will be able to build analog electroni c circuits that could fit in a shoebox and
function according to the same physical principles as a brain. Memristors can potentially learn
like synapses and be used to build human brain-like computers.
Two CMOS circuits connected by a memristor is analogous to two neurons in the brain
connected by a synapse. It is thought that synaptic connections strengthen as the neurons
either side fire and so brain 'circuits' are established which constitutes the basis of human
learning.
Wei Lu, a University of Michigan scientist connected two CMOS circuits by a silver and
silicon Memristor and powered the two CMOS circuits on and off with
v a r yi n g t i m e ga p s between them.
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The memristor alters its state differently depending on the timing of the powering of the
CMOS circuits.
This is said to be the same behavior as that shown by synapses, called "spike
timing plastic dependency", which is thought to be the possible basis for memory and
learning inhuman and other mammalian brains.
The synaptic connection between neurons becomes stronger or weaker, as the time
gap between when they are stimulated becomes shorter or longer. In the same way,
the shorter the time interval the lower the resistance of the memristor to electricity flowing
across it between the two CMOS circuits.
A 20 millisecond time interval between the two CMOS circuits caused a resistance level
roughly half that of a 40 millisecond gap. Lu said: "Cells that fire together wire together... The
memristor mimics synaptic action.
"We show that we can use voltage timing to gradually increase or decrease the electrical
conductance in this memristor-based system. In our brains, similar changes in synapse
conductance essentially give rise to long term memory.
A hybrid circuit—containing many connected memristors and transistors—could help us
research actual brain function and disorders. Such a circuit might even lead to machines that can
recognize patterns the way humans can, in those critical ways computers can’t—for
example, picking a particular face out of a crowd even if it has changed significantly since
our last memory of it.
There are several advantages of the memristor memory over conventional transistorbased memories. One is its strikingly small size. Though memristor is still at its
early development stage, its size is at most one tenths of its RAM counterparts. If the fabrication
technology for memristor is improved, the size and advantage could be even more significant.
Another feature of the memristor is its incomparable potential to store analog information which
enables the memristor to keep multiple bits of information in a memory cell.
Besides these features, the memristor is also an ideal device for implementing synaptic
weights in artificial neural networks.
HP already has plans to implement memristor in a new type of non-volatile memory
which could eventually replace flash and other memory systems.
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Recently, a simple electronic circuit consisting of an LC network and a memristor was
used to model experiments on adaptive behavior of unicellular organisms. It was shown that the
electronic circuit subjected to a train of periodic pulses learns and anticipates the next pulse to
come, similarly to the behavior of slime moulds Physarumpolycephalum subjected to periodic
changes of environment. Such a learning circuit may find applications, e.g., in
pattern recognition.
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CHAPTER 5
CONCLUSION
The reference resistance array-based multilevel memristor memory is
proposed in this paper. The idea has been implemented with two circuits namely the
write-in and the read-out circuits. Simulation of the write-in circuit shows that the
memristors memorize the desired discrete resistance levels regardless of their characteristic
differences. In read-out simulation, contents of the memristors move toward their original values
from the deviated ones whenever the read-out processing is performed.
The proposed multilevel idea of the memristor together with its intrinsic feature of small
size should make the memristor to be a powerful memory device. Also, if the number
of multilevel of memory is increased, the memristor could be an ideal element for synaptic
weight implementation since the synaptic multiplication can be performed simply by Ohm’s law
V=IR in the memristor.
Memristor is the fourth fundamental component. Thus the discovery of a brand new fundamental
circuit element is something not to be taken lightly and has the potential to open the
door to a brand new type of electronics. HP already has plans to implement memristors in a
new type of non-volatile memory which could eventually replace flash and other memory
systems.
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CHAPTER 6
REFERENCES
•H yo n g s u k
Kim
Sah,
M . P . C h a n gj u
Yang
C h u a , L. O . ” M e m r i s t o r
based
m u l t i l e v e l memory” Cellular Nanoscale Networks and Their Applications (CNNA), 2010
12thInternational Workshop, 3-5 Feb. 2010, pp1-6.
•Memristor from Wikipedia, en.wikipedia.org/wiki/Memristor
•http://www.memristor.org/reference/research/13/what-are-memristors
•www.hpl.hp.com/news/2010/apr-jun/
memristor.html
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